Section: New Results

Intuitionistic proofs without syntax

Participant : Lutz Straßburger.

We present Intuitionistic Combinatorial Proofs (ICPs), a concrete geometric semantics of intuitionistic logic based on the principles of classical combinatorial proofs. An ICP naturally factorizes into a linear fragment, a graphical abstraction of an IMLL proof net (an arena net), and a parallel contraction-weakening fragment (a skew fibration). ICPs relate to game semantics, and can be seen as a strategy in a Hyland-Ong arena, generalized from a tree-like to a dag-like strategy. Our first main result, Polynomial Full Completeness, is thatICPs as a semantics are complexity-aware: the translations to and from sequent calculus are size-preserving (up to a polynomial). By contrast, lambda-calculus and game semantics incur an exponential blowup. Our second main result, Local Canonicity, is that ICPs abstract fully and faithfully over the non-duplicating permutations of the sequent calculus. These results have been presented at the LICS 2019 conference [23].